Questions tagged [options]

A contract that gives the owner the right, but not the obligation, to buy or sell a security at a fixed price in the future.

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2
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2answers
277 views

Why represent a digital payoff as a call spread

Pricing a digital caplet using Hull White model, which pays: $1$ if $R>K$, $0$ otherwise. Why would you represent the payoff as a call spread, i.e. $$\text{Payoff} = \frac{(R - (K+\epsilon))^+-(R ...
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1answer
99 views

Single period risk-neutral probability derivation

Let $S_u$ be the price of stock in the up-state one period from now. Let $S_d$ be the price of the stock in the down state. Let $C_u$ be the payoff of a call option at time $1$ in the up-state and ...
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0answers
37 views

How does Intrinsic and Time Premium factor into deep ITM options for leveraged securities

So I'm curious about the downside risk on this trade. Some backstory - I noticed the options chain for TZA had basically no volume or open interest for deep ITM calls about a week ago while also ...
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3answers
114 views

Intepreting European call option when expiration approaches to infinity

Assume that dividend = 0, then the price of call option is $$ C = S\cdot P_{s}[S(T) > K] - e^{-rT}K\cdot P_F[S(T) > K] = SN(d_1)-e^{-rT}KN(d_2) $$ where $P_s[S(T) > K]$ = Probability of ITM ...
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0answers
64 views

How to price barrier options under Black-Scholes?

I am looking for a rigorous proof for the closed form of the price of a barrier option (up-in/up-out) under Black-Scholes model, that is a step by step solution of the solution of the heat equation ...
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93 views

Pricing of strange Asian lookback option with European-style payoff $\max\{ \max_{u\in[0,T]}S_u-\frac1T\sqrt{\int_0^TS_t^2\mathrm{d}t},0\}$

I am trying to price the Asian lookback option at time $t$ with time-$T$ (European) payoff $\max\{M_T-A_T,0\}$, where $$M_t=\max_{u\in[0,t]}S_u,\quad A_t=\frac1t\sqrt{\int_0^tS_u^2\mathrm{d}u},$$ and $...
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73 views

Options Arbitrage

I have a basic question regarding the BSM formula, would be thankful for any assistance. As far as I understand $N(d2)$ and $N(-d2)$ stand for the probability of a Call and Put respectively being ...
4
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1answer
154 views

How much does a rise in volatility in a short-term option affect a longer-term option

How would a rise in implied volatility on a short-term option affect the implied volatility of another short-term option with the same strike, but with slightly-longer expiry? Assuming that the short-...
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0answers
25 views

Latest and currently utilized research on modeling option pricing with IV smile

As per title, where would I find the latest research papers on modeling option pricing, accounting for IV smile? I'm specifically interested in papers that already found practical application in some ...
4
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4answers
172 views

Asymptotics of Call Option as $S\to0$

Let $C(S)$ denote the (initial) value of a call option with underlying spot price $S$. I assume that the underlying has continuous sample paths (not necessarily a geometric Brownian motion though). As ...
2
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1answer
144 views

Aggregate Greeks calculations

My question is simple and slightly amateur but I wanted to get a good foundation on aggregate Greeks calculations. For delta, I understand the delta values for a spread would be first multiplied by ...
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0answers
157 views

Mechanics of index CDS options

I am looking at some documents regarding pricing approaches for index CDS options but none of them give much detail on the mechanics of trading the product. I have looked at the CDX UNTRANCHED ...
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1answer
114 views

Binomial Option pricing, paper by John C. Cox, I don't understand the calculation / choice of u.d.q

[EDIT] Question is answered, just cleaned up some clerical errors in the formulas. [EDIT] Based on the comment I got, let me clarify, I am not stuck on the relationship between the binomial model vs ...
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0answers
54 views

Stock price under Bond numeraire

The Radon-Nikodym derivative going from the bank-acount Numeraire $N(t)$ to the bond numeraire $P(t,T)$ is: $$\frac{dP}{dN}(T|\mathcal{F}_t)=\frac{1}{N(T)P(t,T)}$$ Suppose I now want to price an ...
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2answers
57 views

Equations to Test of local linearity of a derivative security [closed]

Friends any hint as to why is this set of equations a test of linearity of a derivative security? From Taleb - Dynamic Hedging pg. 11 ,, Derivatives are not always ...
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2answers
426 views

Caplet "in arrears" pricing formula

The forward Libor rate $L(t,t_1,t_2)$, with $0 \leq t \leq t_1$, must be a martingale under the T-forward measure associated with the zero coupon bond $P(t,t_2)$ that matures at time $t_2$. Pricing a ...
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1answer
103 views

What is the market standard for measuring historical volatility?

Hope to get some help with the following questions: Can someone explain what is the industry standard to calculate stock options historical volatility? I am using this estimator https://portfolioslab....
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0answers
83 views

Risk neutral probabilities in binomial option pricing with discrete dividends — whose argument is correct?

In trying to discover more about pricing American options with dividend payouts, I found the the post linked here. I notice two disagreeing answers when it comes to determining the replicating ...
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0answers
45 views

Practical Effect of Time-Decay on Variance Swaps?

I want to implement a long vol hedging strategy by rolling spot variance swaps every month. This would be done through replicating spot VIX using the definition of VIX as a portfolio of OTM one-month ...
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0answers
268 views

FX American call option optimal exercise and holding region

Problem I am considering an American call option which gives a domestic investor the right to buy a unit of foreign currency at a strike of $K$ units of domestic currency. I have an exchange rate $S_t$...
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1answer
145 views

Hedging strategy for payoff $\int_0^T\log S_u\mathrm{d}u$

What would a hedging strategy look like for a payoff $\int_0^T\log S_u\mathrm{d}u$? I have determined under Black-Scholes stock dynamics, $$\int_0^T\log S_u\mathrm{d}u=\int_0^t\log S_u\mathrm{d}u+\...
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1answer
239 views

How to price an European put option using binomial model with dividend yield?

The initial stock price (S0) is 45, the stock volatility is 0.20 (20% per annum), and the risk-free rate is 0.02 (2% per annum). Consider a European put option whose strike price is equal to 30, with ...
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1answer
205 views

Meaning of Rebalancing the Gamma in Options?

What does rebalancing the gamma mean? In the Book: Dynamic Hedging at the beginning says: Rebalancing the gamma corresponds to buying and selling the underlying security in order to replicate the ...
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1answer
148 views

Digital call under Ornstein-Uhlenbeck dynamics

I am trying to price a digital option with payoff $\mathbb{I}_{S_T>K}$, where $S_t$ follows the Ornstein-Uhlenbeck dynamics $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma\mathrm{d}W^{\mathbb{Q}}_t$ in the ...
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1answer
119 views

delta neutral option cost

I am trying to understand how an delta neutral profile is generated. I sell a call for strike of 50$ and the delat of this call is 0.5. I buy 0.5*100 stocks to ...
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0answers
61 views

Option where option writer determines type of option to give to holder

I am currently looking at an exotic option that allows the holder, at some time $\tau$, to receive either a call or put — the choice of which is decided by the option writer — of which both have the ...
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1answer
146 views

Pricing of Asian-like option

I am considering an option which has payoff function $\max\{S_T-\frac1\tau\int_0^\tau S_t\mathrm{d}t,0\}$ for a fixed $\tau$ in the risk-neutral measure $\mathrm{d}S_t/S_t=r_t\mathrm{d}t+\sigma_t\...
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1answer
60 views

Improving control variate for variance reduction

I have tried stock price as control variate for my monte carlo simulation, and I am trying to reduce the variance of my estimated price for European Put option. And the code look like this: ...
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0answers
20 views

Example of one-period model that satisfies law of one price but is not free of arbitrage

We know that by the law of one price: in a one-period model $(\overline{\pi},\overline{S})$ for an arbitrage-free market model it follows that for two strategies $\overline{\rho}$ and $\overline{\xi}\...
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1answer
63 views

Low correlation problem in control variate method

I have been trying to use control variate method to reduce the variance of my Monte Carlo Simulation, however, the model is suffering from low correlation problem which makes the control variate ...
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2answers
172 views

Delta hedging for an American call option on a stock with a continuous dividend yield

Let the dividend yield be $\delta$ and $C_u, C_d$ and $S_u, S_d$ be the up and down values for the stock and the call respectively over the period $\Delta t$. In Hull and all other resources I've ...
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3answers
523 views

Trading desk assumes zero percent discount rate?

All the swaption and option models I have encountered at my employer's trading desks have assumed a zero percent discount rate. I have proposed using the LIBOR curve, but management responded that &...
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0answers
111 views

For one-period model, construct a risk-neutral measure $\mathbb P^{*}$ such that the density is constant on $\{S^{1} (<,>,=)c\}$

Consider a one-period arbitrage-free model, it has one risky asset $(\pi^{1},S^{1})$ such that $\pi^{1}>0$, with interest rate on the risk-free asset $(\pi^{0},S^{0})$ at $r > -1$.Furthermore $...
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1answer
102 views

Option Arbitrage Opportunity [closed]

Could you please explain me whether there is an arbitrage opportunity in this situation (added below)?
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1answer
64 views

Cap/Floor on a SpreadOption grid

I have a spread option data from a broker. The rows are the following : STK ATM -0.5 -0.25 ... and the values are forward price ( the strikes used are absolute strike and the value of the raw STK is ...
1
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1answer
891 views

Would it be possible to combine long butterfly with long straddle, achieving profit no matter the outcome?

This has been bugging me for a while, I feel like I'm missing something. Simply put, a long butterfly will make profit if the price at maturity does not change much, as shown below A long straddle is ...
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0answers
54 views

spinoff entity value / adjusted close of a spinoff

When company $A$ spins off company $B$ (i.e. $A = A'+B$), how do we know exactly the adjustment factor of $A'$ and $B$ before trading Note I am not asking to value the new companies. That's a whole ...
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3answers
443 views

Why do transaction costs increase the range of the no-arbitrage bounds for an option's price?

I am reading this book by Mark S. Joshi. Can you help me make sense of one of the exercise questions? Here is the question (from page 40 of the book): Exercise 2.5 Suppose no-arbitrage bounds for an ...
2
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1answer
167 views

Forward starting options concepts

Consider $t_0<t<T$, with $t_0=0$ (today date) and the standard payoff of a vanilla forward starting call option, $F_{t,T} = (S_T - S_t\cdot K)^+$, with strike $K$. If the price of this option is ...
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0answers
48 views

How to derive put option from Black-Scholes equation?

The Question is as follows: The diffusion equation is: I have tried attempting this question by making some change of variables and separating the cumulative distributive function but I get stuck ...
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2answers
57 views

What is IV % actually measuring? [closed]

If the Implied IV of an option is 40%, what is the 40% representing, 40% of what? Does that mean the underlying stock is estimated it may move up or down 40% in a day, month year? The option price may ...
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0answers
33 views

Which volatility should I use in a long-term futures swaption?

Consider an option expiring in 12/31/2023 on an hourly swap from 2024 through 2029 such that: a) I pay the floating price of electricity and b) receive $20 in return. Using shaped monthly futures and ...
2
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1answer
120 views

Covariance matrix for multiple assets - Second attempt

Ok, on the advice of administration I open a new question, hoping that in this way it becomes clearer. Like I said before, I am trying to understand how the authors of this (page 76) and this (page ...
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1answer
46 views

Volatility surface of daily contracts from ATM volatility of quarterly contracts

I'm trying to estimate the volatility surface of an especially illiquid options market; only ATM quotes are available (so Vanna-Volga approximation is not viable) for options on quarterly futures for ...
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0answers
26 views

Choose an investment option only knowing the costs, cost of capital and lifetime

A question in my course hasn't given us much to go on. I'm not sure if there is a specific model I could use to determine which project is the better one. 1st option costs 350.000, has a life ...
4
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4answers
296 views

Confusion about Vega P/L

For someone who has a delta hedged options position, the $\Gamma:= \frac{\partial^2V}{\partial S^2}$ roughly quantifies the amount of money made or lost if $$\frac{1}{\Delta t}\frac{(\Delta S)^2}{S^2} ...
2
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1answer
80 views

European call option on constant volatility or drawn from a volatility distribution

Which is more expensive: A European call option on constant volatility of 30% or or drawn from a random distribution of mean 30%? The answer in A Practical Guide To Quantitative Finance Interviews, ...
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1answer
144 views

Difference b/w Spot Premium and Forward Premium for FX Options

Can someone elaborate the difference between the two, and what is the typical convention used in markets? If there is a mathematical relationship. Any helpful links/guides would be appreciated as well,...
2
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1answer
197 views

Any good papers on Fixed Income Option pricing?

Whilst I have managed to find plenty of material on pricing of Interest Rate Options (i.e. Caps, Floors, Swaptions, spread-options, etc.), I haven't really managed to find any solid papers on the ...
2
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1answer
154 views

Compare equity option volatility under SOFR vs LIBOR

We know that after the big bang from LIBOR to SOFR, LIBOR will eventually disappear. This brings up one question that I do not have a clue to answer: How to evaluate derivative in a consistent manner ...

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